Two hours to White Mountain National Forest, 1.25 to Green Mountain National Forest. Less than 45 minutes to Saint Gaudens Historic Site (a friend’s husband was a ranger there 
).
If I can count the Appalachian Trail, then 20 minutes or so.
Same here, neighbor!
Around 3 hours from the main portion of Everglades National Park. About 2 hours from Ocala National Forest. I’ve never actually visited the Everglades, merely driven through Alligator Alley once or twice. I can’t tell from Google Maps whether that’s technically part of the park but I am not interested enough to find out.
2 hours to Grand Teton
3 hours to Yellowstone
Here is an interesting map to see what is closest to you
Find The Closest National Park With This Handy Map
National Forests are administered by the Dept of Agriculture. National Parks are administered by the Dept of Interior. Very different setups. National Trails (like the AT) are a weird hybrid, with parts of both organizations being involved, but technically they are administered by the National Park Service.
Here’s a map/list of all sites administered by the NPS. There are lots more than the “traditional” parks. Harpers Ferry Center (U.S. National Park Service)
The closest to me would probably be the Minute Man National Historic Park in Lexington. I ride my bike past it often. The JFK National Historic Site in Brookline is slightly further away. The Boston National Historic Park and Boston Harbor Islands National Recreation Area are next. The closest “traditional” national park would be Acadia up in Maine.
My house is 3.5 miles from the Hopewell Culture National Historical Park. Ancient mounds.
As with the many/most other Chicago area Dopers, about an hour from the Indiana Dunes. Designated a National Lakeshore or something, rather than National Park per se. Used to be 20 min away, and walked my dog there daily. That was nice.
Also, I&M National Historic Corridor is less than 20 min away, but that is essentially a bikepath along an old canal.
Several National Battlefields within a 2-2.5 hour drive (Shiloh, Murfeesboro, Chickamauga), but the nearest National Park is Smokey Mountains, about 3-3.5 driving hours up the road.
I’m about 25 miles from the MLK National Historic Site, 35 miles from the Chattahoochee River National Recreation Area (both administered by the National Park Service) and about 45 miles from Kennesaw Mountain National Battlefield Park, and have been all my life. Never been to any of them. In fact, I’ve never been to any of the 12 NPS sites located in Georgia. Yes, that seems odd to me, too. The closest one I’ve actually visited is the Great Smokey Mountain National Park in TN/NC, about 150 miles away.
About 4.5 hours from Crater Lake National Park and about 5 hours to the Redwoods. I’m assuming the OP is not including National Historic sites or monuments.
Bangkok is a couple hours from Khao Yai National Park, Thailand’s first and a Unesco World Heritage Site since 2004. It’s the closest one I can think of offhand.
I live about 2 hours from the Shenandoah National Park, and about 2 blocks from National Park Service land (Grant Circle, in Washington, DC).
Cuyahoga Valley is the closest to me too, about a 3 hr drive.
I just remembered, I am also about 3 blocks from the Lincoln Cottage, a national monument. Incredibly, I’ve never been there, but I keep meaning to go.
Lincoln Cottage in DC? That appears to be part of the National Trust for Historic Preservation, a private non-profit.
I’m about 10 miles from Minute Man Historical National Park, Lowell Historical National Park, and Boston Historical National Park.
I’m a little over an hour east of Rocky Mountain National Park. Lucky me. Sincerely. Very lucky me.
I’m about half an hour from Dartmoor National Park, and we spend a lot of time up there hiking and stuff.
I’m another Cuyahoga Valley-er. I’ve heard that it’s actually the most visited of the national parks, even ahead of Yellowstone, which is no doubt due to it being the closest to a major metropolitan area. It still feels funny to think of it as a “National Park”, though: That’s supposed to be those big places out West in the middle of nowhere. It’s still stuck in my head that it’s a “National Recreational Area”.